afasiask7xg
2022-03-23
Answered

Find the general solution to the following second order differential equations.

$y{}^{\u2033}-5y=0$

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Jambrichp2w2

Answered 2022-03-24
Author has **12** answers

First finding the solutions to the given second-order differential equation by forming the auxiliary equation

Let us take $\frac{dy}{dx}=D$

Then we have,

${D}^{2}-5=0$

${D}^{2}=5$

$D=\pm \sqrt{5}$

We know when the solutions to the auxiliary equation are identical and of opposite sign then the general solution to the given differential equation is

$y={c}_{1}{e}^{at}+{c}_{2}{e}^{bt}$

Here,

$a=\sqrt{5},\text{}b=-\sqrt{5}$

Therefore the general solution to the given differential equation is

$y={c}_{1}{e}^{\sqrt{5}t}+{c}_{2}{e}^{-\sqrt{5}t}$

Jeffrey Jordon

Answered 2022-03-31
Author has **2262** answers

asked 2021-12-31

Find the solution of the following Differential Equations

$(2x+{e}^{y})dx+x{e}^{y}dy=0$

asked 2022-05-31

I am looking to solve the following equations numerically:

$ax=\frac{d}{dt}(f(x,y,t)\frac{dy}{dt}),\phantom{\rule{1em}{0ex}}by=\frac{d}{dt}(g(x,y,t)\frac{dx}{dt})$

For arbitrary functions f and g and constants a and b. I am struggling to find a way to transform this into a system of first order differential equations that I can pass into a solver. It looks like I will need to define these implicitly, but I'm not sure how to do that.

My best attempt so far is the following:

$\begin{array}{rl}{z}_{1}& =f(x,y,t)\frac{dy}{dt}\\ {z}_{2}& =g(x,y,t)\frac{dx}{dt}\\ {z}_{3}& =ax\\ {z}_{4}& =by\end{array}$

${\left(\begin{array}{c}{z}_{1}\\ {z}_{2}\\ {z}_{3}\\ {z}_{4}\end{array}\right)}^{\prime}=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& \frac{a}{g(t,x,y)}& 0& 0\\ \frac{b}{f(t,x,y)}& 0& 0& 0\end{array}\right)\left(\begin{array}{c}{z}_{1}\\ {z}_{2}\\ {z}_{3}\\ {z}_{4}\end{array}\right)$

However, this seems fairly inelegant and assumes that you are always able to divide by f and g. I'm trying to keep this as general as possible, so don't want to make that assumption. Is there a better way to turn this into a system of first order differential equations implicitly?

$ax=\frac{d}{dt}(f(x,y,t)\frac{dy}{dt}),\phantom{\rule{1em}{0ex}}by=\frac{d}{dt}(g(x,y,t)\frac{dx}{dt})$

For arbitrary functions f and g and constants a and b. I am struggling to find a way to transform this into a system of first order differential equations that I can pass into a solver. It looks like I will need to define these implicitly, but I'm not sure how to do that.

My best attempt so far is the following:

$\begin{array}{rl}{z}_{1}& =f(x,y,t)\frac{dy}{dt}\\ {z}_{2}& =g(x,y,t)\frac{dx}{dt}\\ {z}_{3}& =ax\\ {z}_{4}& =by\end{array}$

${\left(\begin{array}{c}{z}_{1}\\ {z}_{2}\\ {z}_{3}\\ {z}_{4}\end{array}\right)}^{\prime}=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& \frac{a}{g(t,x,y)}& 0& 0\\ \frac{b}{f(t,x,y)}& 0& 0& 0\end{array}\right)\left(\begin{array}{c}{z}_{1}\\ {z}_{2}\\ {z}_{3}\\ {z}_{4}\end{array}\right)$

However, this seems fairly inelegant and assumes that you are always able to divide by f and g. I'm trying to keep this as general as possible, so don't want to make that assumption. Is there a better way to turn this into a system of first order differential equations implicitly?

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Use the definition of Laplace Transforms to show that:

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Does ODE initial value problem produce beat or resonance phenomenon?

$x{}^{\u2033}+9x=\mathrm{sin}\left(3t\right)$

$x\left(0\right)={x}^{\prime}\left(0\right)=0$ .

We are allowed to solve differential equations with TI-89.

We are allowed to solve differential equations with TI-89.

asked 2022-01-17

Find the general solution to the differential equation

$\frac{dy}{dt}={t}^{3}+2{t}^{2}-8t$

Also, part 8B. asks: Show that the constant function$y\left(t\right)=0$ is a solution.

Ive

Also, part 8B. asks: Show that the constant function

Ive

asked 2021-09-07

Use Laplace transform to solve the initial-value problem

asked 2022-05-09

Use linear approximation, i.e. the tangent line, to approximate ${11.2}^{2}$ as follows :

Let $f(x)={x}^{2}$ and find the equation of the tangent line to $f(x)$ at $x=11$. Using this, find your approximation for ${11.2}^{2}$.

Let $f(x)={x}^{2}$ and find the equation of the tangent line to $f(x)$ at $x=11$. Using this, find your approximation for ${11.2}^{2}$.